The effect of input cross-spectra on the estimation of frequency response in certain multiple-input systems

The estimation of frequency response of multiple input systems is discussed from the standpoint of systems identification, with application to problems in noise control. The systems considered are assumed to have inputs, either random or deterministic, that are identical in form but shifted in time. Such inputs are found typically as force and pressure excitations in engines, pumps and compressors. It is shown that the input cross-spectra can be neglected for inputs of this type, providing proper frequency smoothing is used. If the time-shift between inputs is not equal or if the analysis bandwidth cannot be chosen arbitrarily, biased estimates of the frequency responses will result when the input cross-spectra are neglected. Expressions for this bias error are developed and several numerical examples are presented showing the effect of analysis bandwidth and timeshift on the bias error. This technique was applied to the problem of estimating the structural-acoustical frequency response of a diesel engine. By neglecting the cross-spectra between the combustion pressures the frequency responses were computed on-line with a small digital processor. As a result experimental and computer time were greatly reduced.